The only observable effect is usually decrease of the delta_cp value (delta_cp being the difference between the heat capacities of the glass and undercooled liquid). Otherwise, the shape of the relaxation overshoot (its relative width and height) and temperature Tg usually remain unchanged. Higher degree of crystallinity may in certain cases slightly shift the overall effect in temperature (with the activation energy remaining the same) - in case of our data the shift was (interestingly) to lower temperatures. For more details you can see: Apparent activation energy of structural relaxation for Se70Te30 glass. R. Svoboda et al., J. Non-Cryst. Solids 356 (2010) 165-168.
The glass transition temperature is a function of chain flexibility and transition occurs when the temp is high enough to allow chain movement/relaxation and therefore the Temp at which this occurs should not be influenced by a small amount of crystallinity. However, the Cp will decrease (compared to a completely amorphous sample) as effectively you are reducing the wt-% of amorphous content in your sample when the network some of the sample is crystalline. As you increase the crystalline content Tg can be unaffected, decrease or increase depending on how the mobility/accessible free-volume is affected by the crystalline content
Yes, with increasing crystallinity the glass transition may become somwhat blurred but I woudl expecta shift in Tg (of polymers) only at very highy crystallinity where the chains belonging to the amorphous phase may become strained.
In that case, I do not see how the transition temperature would be effected if the glass structure is the same. Perhaps there could be a small reduction in activation energy because of nucleation at the interface of the glass/crystalline phase, but even in a completely amorphous material there will be preferred nucleation sites, so this would be very minor. The main effect I would expect is that less heat would be produced during crystallization since part of the sample is already crystallized. Other than I would expect little difference unless the crystalline phase goes through a phase transition. If you look at XRD patterns of the 2 samples and the glass phase has the same short range structure then the crystallization temperature and products should be the same.
I agree with previous answers. In most cases you should not see much change of the Tg. But all of these are applicable for the glasses in which energy needed for crystal growth is higher than Tg. In opposite case you most likely would not see Tg at all. Also, the question about chemical composition of glass and crystals stay open. It is very unlikely to have crystals exactly the same chemical composition as glass and even if it is close near the crystals the composition of the glass would be different than in bulk. This can change Tg as well.